Let's look at context to derive an answer: 1Cor 6:5 I speak to your shame. Is it so, that there is not a wise man among you? no, not one that shall be able to JUDGE between his brethren?
6 But brother goeth to LAW with brother, and that before the unbelievers.
7 Now therefore there is utterly a fault among you, because ye go to LAW one with another. Why do ye not rather take wrong? why do ye not rather suffer yourselves to be defrauded?
8 Nay, ye do wrong, and defraud, and that your brethren.
9 Know ye not that the unrighteous shall not inherit the kingdom of God? Be not deceived: neither fornicators, nor idolaters, nor adulterers, nor effeminate, nor abusers of themselves with mankind,
10 Nor thieves, nor covetous, nor drunkards, nor revilers, nor extortioners, shall inherit the kingdom of God.
11 And such were some of you: but ye are washed, but ye are sanctified, but ye are justified in the name of the Lord Jesus, and by the Spirit of our God.
12 All things are LAWful unto me, but all things are not expedient: all things are LAWful for me, but I will not be brought under the power of any.
Paul is dealing with the church in Corinth who are brothers going to LAW with brother, and that before the unbelievers.
Can a Christian take a fellow brother to judgement before a heathen non-christian, anti-christians judge or judiciary? Yes it is lawful in Rome and most any where to do so.
Is that expedient for the brother or the church? No.
Why, because the judiciary in focus is ungodly, they are not of the kingdom of God. Because this judiciary is of another kingdom with another kingdom law, this judiciary (as in the now no -christian west) are made up of : fornicators, idolaters, adulterers, effeminate, abusers of themselves with mankind, thieves, covetous, drunkards, revilers, extortioners, there don't inherit the kingdom of God.
So why would Christians want to be judged by those who aren't citizens of the kingdom of God?
Not expedient for the Gospel at all.
And again, in context and as a common theme : 1Cor 10:21 Ye cannot drink the cup of the Lord, and the cup of devils: ye cannot be partakers of the Lord's table, and of the table of devils.
22 Do we provoke the Lord to jealousy? are we stronger than he?
23 All things are lawful for me, but all things are not expedient: all things are lawful for me, but all things edify not.
24 Let no man seek his own, but every man another's wealth.
Notice in this is as well dealing with expedience for your brother, even as Chapter 10 continues, it is the Christian's aim to edify his brother according to Paul. So why wouldn't one eat an idols offering even if it is food lawful for a Christian who knows and believes there is only one God and that idols are nothing?
Here is why not: v.33 Even as I please all men in all things, not seeking mine own profit, but the profit of many, that they may be saved.
I hope this Context helps